Hi Rob --
Thanks for the help. The vector form of the equations does speed
things up nicely on these uncoupled equations. What I really want to
do is solve coupled equations (like a spatially-discretized version
of a reaction-diffusion model), which I haven't had as much luck
with. In this case, the vector form you suggested does horribly
compared to the naive form with n[x]'[t]s in it. Any suggestions?
[I know Mathematica can do PDE's directly, but the real problem I
have in mind contains integrals over the spatial variable, so I
thought it would be best to discretize in space manually.]
In[2]:=
tmax=100;
r=1;
k=2;
kp=1;
c=1;
m=0.1;
d=0.001;
xmax=256;
dmat =NDSolve`FiniteDifferenceDerivative[2, Range[0,1,1.0/xmax],
DifferenceOrder\[Rule]2,PeriodicInterpolation\[Rule]True]@
DifferentiationMatrix[];
eqns=Flatten[Table[{
n[x]'[t]\[Equal]
r*n[x][t]*(1-n[x][t]/k)-c*n[x][t]*p[x][t]/(n[x][t]+kp)+
d*(dmat[[Mod[x-1,xmax,1],x]]*n[Mod[x-1,xmax,1]][t]+
dmat[[x,x]]*n[x][t]+
dmat[[Mod[x+1,xmax,1],x]]*n[Mod[x+1,xmax,1]][t]),
p[x]'[t]\[Equal]
c*n[x][t]*p[x][t]/(n[x][t]+kp)-m*p[x][t]+
d*(dmat[[Mod[x-1,xmax,1],x]]*p[Mod[x-1,xmax,1]][t]+
dmat[[x,x]]*p[x][t]+
dmat[[Mod[x+1,xmax,1],x]]*p[Mod[x+1,xmax,1]][t])
},{x,1,xmax}]];
ics=Flatten[
Table[{n[x][0]\[Equal]If[0.4*xmax<x<0.6*xmax,1,0],
p[x][0]\[Equal]0.3},{x,1,xmax}]];
unks=Flatten[Table[{n[x],p[x]},{x,1,xmax}]];
First[Timing[
NDSolve[Flatten[Join[eqns,ics]],unks,{t,0,tmax},MaxSteps\[Rule]
Infinity]
]]
Out[14]=
13.3953 Second
In[15]:=
First[Timing[
NDSolve[{n'[t]\[Equal]r*n[t]*(1-n[t]/k)-c*n[t]*p[t]/(n[t]+kp)
+d*dmat.n[t],
p'[t]\[Equal]c*n[t]*p[t]/(n[t]+kp)-m*p[t]+d*dmat.p[t],
n[0]\[Equal]Table[If[0.4*xmax<x<0.6*xmax,1,0],{x,1,xmax}],
p[0]\[Equal]Table[0.3,{x,1,xmax}]},{n,p},{t,0,tmax},
MaxSteps\[Rule]Infinity]
]]
Out[15]=
65.7573 Second
When diffusion is turned off (d=0), the first method takes 4.5
seconds but the second takes only 1.2 seconds, so I'd like to harness
that speed if possible.
thanks again -- Chris
--
Christopher Klausmeier
Kellogg Biological Station
Michigan State University
Hickory Corners MI 49060
Web: http://www.kbs.msu.edu/Faculty/Klausmeier/Index.htm
Email: klausme1 at msu.edu
Phone: (269) 671-4987